We clarify the differences between Theorem REF  and some of the closely related results.
Wu and Zhu [1]} solved the stability problem for the MHD system with horizontal dissipation
\(\Delta _h u\)  and vertical magnetic diffusion \(\partial _3^2 b\) . It appears that the situation considered
here is more difficult. This is due to the handling of the velocity nonlinearity \((u\cdot \nabla ) u\) .
When the velocity dissipation is only in one direction, the triple-product term
\(((u\cdot \nabla ) u, u)_{H^3}\)  is much more difficult to control than any triple product terms
generated for the MHD system considered in [1]}. In fact, this term is exactly the reason
why the well-posedness problem on the 3D anisotropic Navier-Stokes (REF ) is open.
One main contribution of this paper is the handling of the Navier-Stokes nonlinearity
when the dissipation of the velocity is only in a single direction. The smoothing and stabilizing effect of the magnetic field on the fluids, and the elaborate construction of
time-weighted energy functional are the key ingredients of this successful story. We remark that
there is a very large mathematical literature on the incompressible MHD equations. In particular, there have been substantial recent developments on the well-posedness and stability problems, and
significant progress has been made (see, e.g., [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]}, [16]}, [17]}, [18]}, [19]}, [20]}, [21]}, [22]}, [23]}, [24]}, [25]}, [26]}, [27]}, [28]}, [29]}, [30]}, [31]}, [32]}, [33]}, [34]}, [35]}, [36]}, [37]}, [38]}, [39]}, [40]}, [41]}, [42]}, [43]}, [44]}, [45]}, [46]}, [47]}, [48]}, [49]}, [50]}, [51]}, [52]}, [53]}, [54]}, [55]}).
